Math 2374 Spring 2006
Midterm 2 Solution  Page 1 of 5
March 29, 2006
1. (25 points) Let the curve
C
in the (
x, y
)plane be the boundary of the unit square:
C
consists
of four line segments, from (0
,
0) to (1
,
0); from (1
,
0) to (1
,
1); from (1
,
1) to (0
,
1); and from
(0
,
1) to (0
,
0). Evaluate the line integral
C
xy
(

1 +
x
2
+ 9)
dx
+
1
3
(
x
2
+ 9)
3
/
2
dy
by using Green’s Theorem.
Solution.
Need to observe first that
C
is a closed curve which is the boundary of the unit square
D
:
0
x
1, 0
y
1. Since it is closed we may use Green’s theorem:
C
P dx
+
Q dy
=
D
∂
Q
∂
x

∂
P
∂
y
dxdy
In our case
P
=
xy
(
x
2
+ 9

1),
∂
P
∂
y
=
x
(
x
2
+ 9

1) =
x
x
2
+ 9

x
Q
=
1
3
(
x
2
+ 9)
3
/
2
,
∂
Q
∂
x
=
1
3
3
2
2
x
(
x
2
+ 9)
1
/
2
=
x
x
2
+ 9
Thus
∂
Q
∂
x

∂
P
∂
y
=
x
x
2
+ 9

(
x
x
2
+ 9

x
) =
x
By Green’s theorem
C
P dx
+
Q dy
=
D
x dxdy
=
1
0
1
0
x dy dx
=
1
0
x dx
= 1
/
2
.
Deductions. Mistake in the computation of the last integral 2 pts o
ff
. Wrong computation of
∂
Q
∂
x

∂
P
∂
y
leads to 57 pts o
ff
provided that integration after still done up to a ”right” number.
If not, other deductions might take place.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Math 2374 Spring 2006
Midterm 2 Solution  Page 2 of 5
March 29, 2006
2. (25 points) You build a fence so that the base of the fence is the circle
C
given by
x
2
+
y
2
= 4.
If the height of the fence at any point (
x, y
) along the circle is given by
f
(
x, y
) =
x
+4, calculate
the area of the fence.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 Mosher
 Math, Cos, dx dy

Click to edit the document details